Remark 1.2 
The corresponding question for the quasi-geostrophic equation
has been a focus of significant effort (see e.g. [1]}, [2]}, [3]}, [4]}, [5]}, [6]}, [7]})
and the critical Q-G equation has been recently resolved in [5]} for periodic case.
Based on [5]}, Abidi-Hmidi in [1]} and Dong-Du in [4]}
give the corresponding result for Cauchy problem of the critical Q-G equation in the framework of Besov space
and Sobolev space, respectively.
After the present paper is completed, Prof. J.Wu and H.Dong informed us that the authors in [12]}
gave the global well-posedness for the critical fractal Burgers equation in inhomogeneous space \(H^{\frac{1}{2}}\) 
by similar argument in [4]}.

